2.8 remove doubles has changed to Merge by Distance
As of writing this answer, the option is not present in Blender, or at least not searchable through F3 and buried somewhere deep. What works instead, is merging by distance by either of these methods:
(remember to first select at least two vertices: at least one will be kept, while those, that are too close to kept vertices (depending on the setting), will be removed)

From the Mesh menu select Clean up and then Merge by distance.
[No longer in 2.90.1] From the Vertex menu select Merge Vertices and then by distance.
From the context menu (right mouse button by default) select Merge Vertices and then by distance.
Press M, then B keys. [before 2.90.1 Alt+M, then B keys.]

A panel with settings will appear on bottom left:

After clicking it, it will expand:

On bottom you should see the number of removed vertices, depending on current setting:

Step-by-step example

Press CTRL+N, then G for a new general file.
Make sure the cube is selected.
Press TAB to enter edit mode
Press ALT+A to deselect all vertices.
Click on a vertex.
Press G twice, and slide the vertex as far to another vertex as possible, so they both occupy the same position, confirm by clicking.
Press A to select all vertices (to make sure both vertices in the same spot are selected)
Press M, then B to merge by distance. Regardless of the setting, at least one vertex will be removed which should be confirmed by a status bar message: 
Press ALT+A to deselect all vertices again.
Click on the vertex that had a &quot;duplicate&quot; on the same position.
Either press G to start moving the vertex and see all edges are connected to just this one, and the other is gone or H to hide the vertex and see there's no second vertex at the same spot.


Beware, that this will remove not only vertices that have the exact same coordinates, but also those that are within 0.0001m of each-other. You can decrease this value to as low as 0.000001m.
Thanks to the comments from Hikariztw and JtheNinja, I investigated the issue and indeed the old remove doubles worked exactly the same.
In Blender 2.79 the merge distance seemed to default to zero:

but after clicking the input field it would reveal that the 0.000 was just a rounded view of an actual non-zero value:

In both cases (Blender 2.79 and Blender 2.8) the default value is 0.0001m and the minimum value is 0.000001m (in Blender 2.79 you have to go to scene settings and set metric system for length in order to see 1 micrometer instead of 0.000001 without a unit).